Home Robotics C++ Physics II AP Physics B Electronics AP Java Astronomy Independent Study Summer Session Contests  About
                                                       

Energy in the Simple Harmonic Oscillator

 

Note the technique (common) for solution

 

1.  Analyze at extreme points and in between

2.  Apply conservation law(s)

 

Ø PE = elastic potential energy of spring

         PE = (1/2)kx2

Ø E = total mechanical energy of a mass-spring system = KE + PE

         E = (1/2)mv2 + (1/2)kx2

 

Ø With no friction, the total mechanical energy remains constant


Ø At extreme points, there is no motion and all of the energy is potential

         E = (1/2)m(0)2 + (1/2)kA2 = (1/2)kA2

 

Ø  At the equilibrium point, all of the energy is kinetic

         E = (1/2)mv2max + (1/2)k(0)2 = (1/2)mv2max

Ø  Since energy is conserved throughout

          E = (1/2)mv2 + (1/2)kx2 = (1/2)kA2


Ø  From the 2 equations in blue Italics above

 

         (1/2)mv2max = (1/2)kA2  ð mv2max = kA2 ð

 

         v2max =  (k/m)/A2

 

Ø  Finally, from the conservation of total mechanical energy (second equation from top above)

 

         (1/2)mv2 + (1/2)kx2 = (1/2)KA2   and, since we showed that v2max =  (k/m)/A2 (previous step above)

 

         v =   ± vmax(1 - x2/A2)1/2